"I think John is doing the right thing," Trump added.
Trump's remarks came in a wide-ranging interview at Trump Tower the day after he notched a crushing victory in the Indiana GOP primary, flattening the final roadblock in his meteoric 10-month rise from celebrity candidate to presumptive Republican nominee.
Trump shared his views on a range of domestic and foreign policy issues, saying he'd consider raising the minimum wage. He addressed some of the controversial proposals that have blighted his favorability among a majority of Americans, and said "of course" he didn't believe Ted Cruz's father was linked to the JFK assassination despite circulating a tabloid report just one day earlier.
To mark that beginning, Trump's campaign is beginning to compile a shortlist of potential running mates, and the real estate mogul said he is forming a committee to vet potential picks -- a group that he said will likely include his former rivals-turned-supporters Ben Carson and New Jersey Gov. Chris Christie.
Although Kasich has been critical of Trump's rhetoric and many of his proposals on the campaign trail, Trump insisted that he has "a very good relationship with John."
"I think John will be very helpful with Ohio," Trump said of Kasich and his home state, which will once again be a battleground state in November's general election.
Democratic presidential front-runner Hillary Clinton, meanwhile, has yet to dispatch her rival for the Democratic nomination, Vermont Sen. Bernie Sanders.
"There's been a little flip," Trump said. "I thought that I'd be going longer and she'd be going shorter."
"I thought that I'd be out there and she'd be campaigning against me," Trump said, noting that he will now be in a position to begin campaigning against the likely Democratic nominee while Clinton continues her contest against Sanders.
The general election, Trump said, has "essentially" begun.
'Looking at' raising minimum wage
Trump, speaking with CNN for the first time as the presumptive GOP nominee, offered a glimpse of how he will seek to expand his appeal beyond the Republican primary electorate to whom he has catered his campaign platform.
Asked whether he would consider raising the minimum wage, Trump said he is "looking at that" and "open to doing something with it," despite claiming months earlier that he opposed raising the minimum wage because the U.S. needs to keep wages low
to compete with foreign countries.
"I'm actually very different from most Republicans" on the issue, Trump said in the interview Wednesday.
Republican lawmakers are overwhelmingly opposed to raising the federal minimum wage, which is currently $7.25 per hour, while Democrats have called for raising the wage floor to as high as $15 per hour.
Trump took fire from conservatives and Republican primary opponents earlier in the campaign when he told Fox News that "everything is negotiable" as he responded to a report that he suggested to The New York Times editorial board in an off-the-record meeting that his immigration views are in fact more flexible
than the hardline positions he laid out during his primary campaign.
Trump and his aides have also suggested that he would begin acting more "presidential" after he defeats his primary opponents -- but Trump previewed few changes in that regard Wednesday.
"I don't like to change," Trump said, noting that he already feels like he is "a presidential person."
Trump said the tone of his general election campaign -- most likely against front-runner Clinton -- would hinge on the tone the Democrat's campaign sets.
"A lot of that's going to depend on how they treat me," Trump said. "I don't really like hitting people first."
While Trump predicted that he will put traditionally blue states like New York and Michigan into contention and "easily" defeat Clinton for the presidency, Trump conceded that Clinton will be a formidable debate opponent.
Fighting back against Cruz
Trump certainly did not mollify his tone or blunt his attacks as voters headed to the polls in Indiana -- now his final contested primary election -- when he suggested Tuesday morning on Fox News that Cruz's father was somehow involved with Lee Harvey Oswald, the suspected assassin of President John F. Kennedy. The baseless accusation had been lifted from the pages of the supermarket tabloid National Enquirer.
Trump said Wednesday that he "of course" does not believe the allegation to be true, but insisted he was fighting back against "horrible statements" the elder Cruz made about him.
"I don't believe it, but I did say let people read it," Trump said.
Trump would not say whether he was now held to a higher standard as the presumptive GOP nominee, but insisted that he "wasn't at the time."
The system: still rigged
Trump's conquest over a once seemingly impossible path to the Republican nomination did not keep the billionaire from continuing to rail against the Republican nominating process.
Asked if he still believed he had faced a rigged system, Trump did not hesitate.
"100%," he said. "The only way I got it, I went for the knockout."
Trump, who said he believes he can win over many Sanders supporters, also knocked the Democratic system -- which relies not just on voting but also unbound superdelegates -- as unfair and said he is glad he has been able to "point out some real weaknesses" in the primary process.
But he also reflected on how far he has come, noting that he had to defeat the large field of "very capable people" and that he "gave up a lot" to run for president.
And he said he believes he knew he could win all along: "Subconsciously, I must've thought I was going to win."
Reiterates support for Muslim ban
Trump also weighed in on pressing foreign policy issues, including aggressive moves Russian warplanes have made recently near U.S. naval ships and how his proposal to temporarily ban Muslims would impact U.S. relations with Muslim nations.
Trump said he would be able to call Russian President Vladimir Putin and get him to keep his warplanes far from American military ships by simply saying: "Don't do it again."
"I think I will have a good relationship with him," Trump said of Putin, the Russian autocrat who in December called Trump "talented" and "absolutely the leader in the presidential race."
Trump did not back away from his proposal to temporarily ban all foreign Muslims from entering the U.S., but said he would work with Muslim countries in the fight against terrorism -- but put the onus on those countries first.
"They have to save themselves, and we can help them, but they have to save themselves," Trump said.
Trump also argued against a proposal for the federal government to bail out Puerto Rico, the U.S. territory that is suffering a crippling debt crisis.
While touting his ability to flip indebted ventures into profitability and to emerge unscathed from his companies' bankruptcies, Trump said Puerto Rico needs to "cut that debt way down" without the U.S. federal government's help.
"I'm the king of debt. I love debt," Trump said. "The problem with Puerto Rico is they have far too much debt."
Improving the GOP convention
Trump no longer has to plan for the bruising prospect of a contested convention, but the now-presumptive nominee said he intends to inject his flair for show business into the party's summer convention in Cleveland.
Knocking the 2012 convention as the "lowest-rated convention in the history of conventions," Trump said he would "like to add something that would make people happy."
"We have to promote our country," Trump said. "People are very depressed."
But Trump avoided getting wrapped up in the flash he plans to add to the convention.
"Much more important than exciting," Trump said, "we want to have good ideas."